3.74.71 \(\int \frac {\log ^{2 x^2}(x) (5 x \log (2) \log ^2(3)+10 x \log (2) \log ^2(3) \log (x) \log (\log (x)))}{32 \log (x)+192 \log (3) \log ^{1+x^2}(x)+384 \log ^2(3) \log ^{1+2 x^2}(x)+256 \log ^3(3) \log ^{1+3 x^2}(x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {5 \log (2)}{16 \left (4+\frac {2 \log ^{-x^2}(x)}{\log (3)}\right )^2} \]

________________________________________________________________________________________

Rubi [F]  time = 0.99, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\log ^{2 x^2}(x) \left (5 x \log (2) \log ^2(3)+10 x \log (2) \log ^2(3) \log (x) \log (\log (x))\right )}{32 \log (x)+192 \log (3) \log ^{1+x^2}(x)+384 \log ^2(3) \log ^{1+2 x^2}(x)+256 \log ^3(3) \log ^{1+3 x^2}(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Log[x]^(2*x^2)*(5*x*Log[2]*Log[3]^2 + 10*x*Log[2]*Log[3]^2*Log[x]*Log[Log[x]]))/(32*Log[x] + 192*Log[3]*L
og[x]^(1 + x^2) + 384*Log[3]^2*Log[x]^(1 + 2*x^2) + 256*Log[3]^3*Log[x]^(1 + 3*x^2)),x]

[Out]

(5*Log[2]*Log[3]^2*Defer[Int][(x*Log[x]^(-1 + 2*x^2))/(1 + Log[9]*Log[x]^x^2)^3, x])/32 + (5*Log[2]*Log[3]^2*D
efer[Int][(x*Log[x]^(2*x^2)*Log[Log[x]])/(1 + Log[9]*Log[x]^x^2)^3, x])/16

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 x \log (2) \log ^2(3) \log ^{-1+2 x^2}(x) (1+2 \log (x) \log (\log (x)))}{32 \left (1+\log (9) \log ^{x^2}(x)\right )^3} \, dx\\ &=\frac {1}{32} \left (5 \log (2) \log ^2(3)\right ) \int \frac {x \log ^{-1+2 x^2}(x) (1+2 \log (x) \log (\log (x)))}{\left (1+\log (9) \log ^{x^2}(x)\right )^3} \, dx\\ &=\frac {1}{32} \left (5 \log (2) \log ^2(3)\right ) \int \left (\frac {x \log ^{-1+2 x^2}(x)}{\left (1+\log (9) \log ^{x^2}(x)\right )^3}+\frac {2 x \log ^{2 x^2}(x) \log (\log (x))}{\left (1+\log (9) \log ^{x^2}(x)\right )^3}\right ) \, dx\\ &=\frac {1}{32} \left (5 \log (2) \log ^2(3)\right ) \int \frac {x \log ^{-1+2 x^2}(x)}{\left (1+\log (9) \log ^{x^2}(x)\right )^3} \, dx+\frac {1}{16} \left (5 \log (2) \log ^2(3)\right ) \int \frac {x \log ^{2 x^2}(x) \log (\log (x))}{\left (1+\log (9) \log ^{x^2}(x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 38, normalized size = 1.58 \begin {gather*} -\frac {5 \log (2) \log ^2(3) \left (1+\log (81) \log ^{x^2}(x)\right )}{64 \log ^2(9) \left (1+\log (9) \log ^{x^2}(x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Log[x]^(2*x^2)*(5*x*Log[2]*Log[3]^2 + 10*x*Log[2]*Log[3]^2*Log[x]*Log[Log[x]]))/(32*Log[x] + 192*Lo
g[3]*Log[x]^(1 + x^2) + 384*Log[3]^2*Log[x]^(1 + 2*x^2) + 256*Log[3]^3*Log[x]^(1 + 3*x^2)),x]

[Out]

(-5*Log[2]*Log[3]^2*(1 + Log[81]*Log[x]^x^2))/(64*Log[9]^2*(1 + Log[9]*Log[x]^x^2)^2)

________________________________________________________________________________________

fricas [B]  time = 0.61, size = 45, normalized size = 1.88 \begin {gather*} -\frac {5 \, {\left (4 \, \log \relax (x)^{\left (x^{2}\right )} \log \relax (3) \log \relax (2) + \log \relax (2)\right )}}{256 \, {\left (4 \, \log \relax (x)^{2 \, x^{2}} \log \relax (3)^{2} + 4 \, \log \relax (x)^{\left (x^{2}\right )} \log \relax (3) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x*log(2)*log(3)^2*log(x)*log(log(x))+5*x*log(2)*log(3)^2)*exp(x^2*log(log(x)))^2/(256*log(3)^3*l
og(x)*exp(x^2*log(log(x)))^3+384*log(3)^2*log(x)*exp(x^2*log(log(x)))^2+192*log(3)*log(x)*exp(x^2*log(log(x)))
+32*log(x)),x, algorithm="fricas")

[Out]

-5/256*(4*log(x)^(x^2)*log(3)*log(2) + log(2))/(4*log(x)^(2*x^2)*log(3)^2 + 4*log(x)^(x^2)*log(3) + 1)

________________________________________________________________________________________

giac [B]  time = 1.06, size = 73, normalized size = 3.04 \begin {gather*} -\frac {5 \, \log \relax (x)^{\left (x^{2}\right )} \log \relax (3) \log \relax (2)}{64 \, {\left (4 \, \log \relax (x)^{2 \, x^{2}} \log \relax (3)^{2} + 4 \, \log \relax (x)^{\left (x^{2}\right )} \log \relax (3) + 1\right )}} - \frac {5 \, \log \relax (2)}{256 \, {\left (4 \, \log \relax (x)^{2 \, x^{2}} \log \relax (3)^{2} + 4 \, \log \relax (x)^{\left (x^{2}\right )} \log \relax (3) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x*log(2)*log(3)^2*log(x)*log(log(x))+5*x*log(2)*log(3)^2)*exp(x^2*log(log(x)))^2/(256*log(3)^3*l
og(x)*exp(x^2*log(log(x)))^3+384*log(3)^2*log(x)*exp(x^2*log(log(x)))^2+192*log(3)*log(x)*exp(x^2*log(log(x)))
+32*log(x)),x, algorithm="giac")

[Out]

-5/64*log(x)^(x^2)*log(3)*log(2)/(4*log(x)^(2*x^2)*log(3)^2 + 4*log(x)^(x^2)*log(3) + 1) - 5/256*log(2)/(4*log
(x)^(2*x^2)*log(3)^2 + 4*log(x)^(x^2)*log(3) + 1)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 31, normalized size = 1.29

method	result	size
risch	\(-\frac {5 \ln \relax (2) \left (4 \ln \relax (3) \ln \relax (x )^{x^{2}}+1\right )}{256 \left (2 \ln \relax (3) \ln \relax (x )^{x^{2}}+1\right )^{2}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x*ln(2)*ln(3)^2*ln(x)*ln(ln(x))+5*x*ln(2)*ln(3)^2)*exp(x^2*ln(ln(x)))^2/(256*ln(3)^3*ln(x)*exp(x^2*ln(
ln(x)))^3+384*ln(3)^2*ln(x)*exp(x^2*ln(ln(x)))^2+192*ln(3)*ln(x)*exp(x^2*ln(ln(x)))+32*ln(x)),x,method=_RETURN
VERBOSE)

[Out]

-5/256*ln(2)*(4*ln(3)*ln(x)^(x^2)+1)/(2*ln(3)*ln(x)^(x^2)+1)^2

________________________________________________________________________________________

maxima [B]  time = 0.51, size = 45, normalized size = 1.88 \begin {gather*} -\frac {5 \, {\left (4 \, \log \relax (x)^{\left (x^{2}\right )} \log \relax (3) \log \relax (2) + \log \relax (2)\right )}}{256 \, {\left (4 \, \log \relax (x)^{2 \, x^{2}} \log \relax (3)^{2} + 4 \, \log \relax (x)^{\left (x^{2}\right )} \log \relax (3) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x*log(2)*log(3)^2*log(x)*log(log(x))+5*x*log(2)*log(3)^2)*exp(x^2*log(log(x)))^2/(256*log(3)^3*l
og(x)*exp(x^2*log(log(x)))^3+384*log(3)^2*log(x)*exp(x^2*log(log(x)))^2+192*log(3)*log(x)*exp(x^2*log(log(x)))
+32*log(x)),x, algorithm="maxima")

[Out]

-5/256*(4*log(x)^(x^2)*log(3)*log(2) + log(2))/(4*log(x)^(2*x^2)*log(3)^2 + 4*log(x)^(x^2)*log(3) + 1)

________________________________________________________________________________________

mupad [B]  time = 5.79, size = 30, normalized size = 1.25 \begin {gather*} -\frac {5\,\ln \relax (2)\,\left (4\,\ln \relax (3)\,{\ln \relax (x)}^{x^2}+1\right )}{256\,{\left (2\,\ln \relax (3)\,{\ln \relax (x)}^{x^2}+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x^2*log(log(x)))*(5*x*log(2)*log(3)^2 + 10*x*log(log(x))*log(2)*log(3)^2*log(x)))/(32*log(x) + 192*
exp(x^2*log(log(x)))*log(3)*log(x) + 384*exp(2*x^2*log(log(x)))*log(3)^2*log(x) + 256*exp(3*x^2*log(log(x)))*l
og(3)^3*log(x)),x)

[Out]

-(5*log(2)*(4*log(3)*log(x)^(x^2) + 1))/(256*(2*log(3)*log(x)^(x^2) + 1)^2)

________________________________________________________________________________________

sympy [A]  time = 0.40, size = 60, normalized size = 2.50 \begin {gather*} \frac {- 20 e^{x^{2} \log {\left (\log {\relax (x )} \right )}} \log {\relax (2 )} \log {\relax (3 )} - 5 \log {\relax (2 )}}{1024 e^{2 x^{2} \log {\left (\log {\relax (x )} \right )}} \log {\relax (3 )}^{2} + 1024 e^{x^{2} \log {\left (\log {\relax (x )} \right )}} \log {\relax (3 )} + 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x*ln(2)*ln(3)**2*ln(x)*ln(ln(x))+5*x*ln(2)*ln(3)**2)*exp(x**2*ln(ln(x)))**2/(256*ln(3)**3*ln(x)*
exp(x**2*ln(ln(x)))**3+384*ln(3)**2*ln(x)*exp(x**2*ln(ln(x)))**2+192*ln(3)*ln(x)*exp(x**2*ln(ln(x)))+32*ln(x))
,x)

[Out]

(-20*exp(x**2*log(log(x)))*log(2)*log(3) - 5*log(2))/(1024*exp(2*x**2*log(log(x)))*log(3)**2 + 1024*exp(x**2*l
og(log(x)))*log(3) + 256)

________________________________________________________________________________________